Monotone Sequence¶

Definition¶

A sequence $(a_n)_{n=1}^\infty$ is (monotonically) increasing or nondecreasing if $a_{n+1}\geq a_n$ for all $n\geq 1\text{,}$ and it is (monotonically) decreasing or nonincreasing if $a_n\geq a_{n+1}$ for all $n\geq 1\text{.}$

A sequence $(a_n)_{n=1}^\infty$ is strictly increasing if $a_{n+1}> a_n$ for all $n\geq 1\text{,}$ and it is strictly decreasing if $a_n> a_{n+1}$ for all $n\geq 1\text{.}$

Interpretation¶

A monotone sequence moves in one direction only: an increasing sequence never steps down, and a decreasing sequence never steps up. Equal consecutive terms are allowed unless the word "strictly" is added.

Two sequences of points: an increasing one rising with some equal steps, and a decreasing one falling.

Neither sequence in the figure is strictly monotone. The blue increasing sequence has two flat steps, $a_3=a_4$ and $a_7=a_8$ $a_{8}$ , and the red decreasing sequence has one, $a_5=a_6$ $a_{6}$ . A single step in the wrong direction, however small, would make a sequence non-monotone.

The definition compares consecutive terms only. Applying transitivity of the inequalities repeatedly gives $a_m\le a_n$ for all $m\le n$ in the increasing case. To check monotonicity, one usually studies the sign of $a_{n+1}-a_n$ $a_{n}$ : for $a_n=\frac{n}{n+1}$ the difference is $\frac{1}{(n+1)(n+2)}>0$ $0$ , so the sequence is strictly increasing. A constant sequence is both increasing and decreasing.

Monotonicity is the hypothesis that drives the monotone convergence theorem: an increasing sequence converges if and only if it is bounded above. The indices $n_1<n_2<\cdots$ of a subsequence form a strictly increasing sequence. The same words return for functions as monotone functions.

A common slip is to assume that a convergent sequence is monotone. The sequence $\frac{(-1)^n}{n}$ converges to $0$ but alternates in sign, so it is neither increasing nor decreasing.

Key Results¶

Monotone sequences may converge or diverge.
Bounded monotone sequences are convergent.

Examples¶

$a_n=\dfrac{1}{n}$ is decreasing and convergent.
$a_n=n$ is increasing and divergent.

Prerequisites¶

Inequality

Sources¶

Math I: Calculus · Lecture 3

Aliases: monotonic sequence Keywords: increasing sequence, decreasing sequence, nondecreasing sequence, nonincreasing sequence Symbols: infinity

Last updated: 20 October 2026 03:02